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\author{Class 2019 Math and Applied Math }
\title{Applied stochastic processes - Homework 04}
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%\date{2021 年 2 月 28 日}
\date{April 13, 2021}
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%\subsection{Homework 04}
%E5.1.1, E5.1.7, P5.1.9, E5.2.1, E5.2.3, P5.2.1.

\begin{document}

\maketitle

\begin{enumerate}

\item [E5.1.1.] Defects occur along the length of a filament at a rate of $\lambda = 2$ per foot.
\begin{enumerate}
\item  Calculate the probability that there are no defects in the first foot of the filament.
\item  Calculate the conditional probability that there are no defects in the second foot of the filament, given that the first foot contained a single defect.
\end{enumerate}


\item [E5.1.7.] Suppose that customers arrive at a facility according to a Poisson process having rate $\lambda = 2$. 
Let $X(t)$ be the number of customers that have arrived up to time $t$. 
Determine the following probabilities and conditional probabilities:
\begin{enumerate}
\item  $\mathbb{P}\{X(1)=2\}$.
\item  $\mathbb{P}\{X(1) = 2 \text{ and } X(3) = 6\}$. 
\item  $\mathbb{P}\{X(1)=2 \mid X(3)=6\}$.
\item  $\mathbb{P}\{X(3)=6 \mid X(1)=2\}$.
\end{enumerate}


\item [P5.1.9.] Arrivals of passengers at a bus stop form a Poisson process $X(t)$ with rate $\lambda=2$ per unit time. 
Assume that a bus departed at time $t = 0$ leaving no customers behind. 
Let $T$ denote the arrival time of the next bus. Then, the number of passengers present when it arrives is $X(T)$. 
Suppose that the bus arrival time $T$ is independent of the Poisson process and that $T$ has the uniform probability density function $f_T(t) = 1$ for $0\le t\le 1$, and $f_T(t)=0$ elsewhere.
\begin{enumerate}
\item  Determine the conditional moments $\mathbb{E} [ X(T) \mid T = t ]$ and $\mathbb{E} [ X(T)^2 \mid T = t ]$.
\item  Determine the mean $\mathbb{E} [ X(T) ]$ and variance $\mathbb{V}ar [ X(T) ]$.
\end{enumerate}


\item [E5.2.1.] Determine numerical values to three decimal places for $\mathbb{P}\{X = k\}, k = 0, 1, 2$, when
\begin{enumerate}
\item  $X$ has a binomial distribution with parameters $n = 20$ and $p = 0.06$. 
\item  $X$ has a binomial distribution with parameters $n = 40$ and $p = 0.03$.
\item  $X$ has a Poisson distribution with parameter $\lambda = 1.2$.
\end{enumerate}


\item [E5.2.3.] A large number of distinct pairs of socks are in a drawer, all mixed up. A small number of individual socks are removed. Explain in general terms why it might be plausible to assume that the number of pairs among the socks removed might follow a Poisson distribution.


\item [P5.2.1.] Let $X(n, p)$ have a binomial distribution with parameters $n$ and $p$. 
Let $n \to\infty$ and $p \to 0$ in such a way that $np = \lambda$. Show that
\begin{eqnarray*}
\lim\limits_{n\to\infty} \mathbb{P}\{X(n,p)=0\}=e^{-\lambda}, 
\text{ and }
\lim\limits_{n\to\infty} \frac{ \mathbb{P}\{X(n,p)=k+1 \} } { \mathbb{P}\{X(n,p)=k \} } = \frac{\lambda}{k+1} \text{ for } k=0,1,\cdots. 
\end{eqnarray*}


\end{enumerate}


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\subsection{Homework 01}
E3.1.2, P3.1.4, E3.2.2, P3.2.4, E3.3.2, P3.3.6.

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\subsection{Homework 02}
E.3.4.1, E3.4.2, P3.4.1, P3.4.5, E3.5.1, P3.5.1. 

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\subsection{Homework 03}
E4.1.10, P4.1.1, P4.1.5, E4.3.1, E4.3.2, E4.4.2.

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\subsection{Homework 04}
E5.1.1, E5.1.7, P5.1.10, E5.2.1, P5.2.1.

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\subsection{Homework 05}
E5.3.1, E5.3.3, E5.3.7, P5.3.1, E5.4.1, E5.4.3. 

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\subsection{Homework 06}
E6.1.1, E6.1.2, P6.1.1, P6.1.2.

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\subsection{Homework 07}
E7.1.2, E7.1.3, E7.2.1, E7.2.3, P7.2.1.

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\subsection{Homework 08}
E8.1.1, E8.1.2, E8.1.4, P8.1.1, P8.1.3, E8.2.1.

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